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Research Papers

On the Solutions of Interval Systems for Under-Constrained and Redundant Parallel Manipulators

[+] Author and Article Information
Leila Notash

Mem. ASME
Department of Mechanical and
Materials Engineering,
Queen's University,
Kingston, ON K7L 3N6, Canada
e-mail: leila.notash@queensu.ca

Manuscript received April 12, 2017; final manuscript received August 23, 2017; published online September 26, 2017. Assoc. Editor: Raffaele Di Gregorio.

J. Mechanisms Robotics 9(6), 061009 (Sep 26, 2017) (9 pages) Paper No: JMR-17-1107; doi: 10.1115/1.4037803 History: Received April 12, 2017; Revised August 23, 2017

For under-constrained and redundant parallel manipulators, the actuator inputs are studied with bounded variations in parameters and data. Problem is formulated within the context of force analysis. Discrete and analytical methods for interval linear systems are presented, categorized, and implemented to identify the solution set, as well as the minimum 2-norm least-squares solution set. The notions of parameter dependency and solution subsets are considered. The hyperplanes that bound the solution in each orthant characterize the solution set of manipulators. While the parameterized form of the interval entries of the Jacobian matrix and wrench produce the minimum 2-norm least-squares solution for the under-constrained and over-constrained systems of real matrices and vectors within the interval Jacobian matrix and wrench vector, respectively. Example manipulators are used to present the application of methods for identifying the solution and minimum norm solution sets for actuator forces/torques.

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References

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Notash, L. , 2015, “ Analytical Methods for Solution Sets of Interval Wrench,” ASME Paper No. DETC2015-47575.
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Notash, L. , 2016, “ On the Solution Set for Positive Wire Tension With Uncertainty in Wire-Actuated Parallel Manipulators,” ASME J. Mech. Rob., 8(4), p. 044506. [CrossRef]
Notash, L. , 2017, “ Wrench Accuracy for Parallel Manipulators and Interval Dependency,” ASME J. Mech. Rob., 9(1), p. 011008. [CrossRef]
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Notash, L. , 2017, “ Solutions of Interval Systems for Under-Constrained and Redundant Parallel Manipulators,” ASME Paper No. DETC2017-67091.
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Figures

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Fig. 1

Parallel manipulators with (a) two actuators and (b) three actuators

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Fig. 2

Enclosure and subsets of solution set

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Fig. 3

Parameters of planar manipulators

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Fig. 4

2-norm of midpoint and radius of residual vector: (a) in solution (b) outside solution set

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Fig. 5

Solution and minimum norm solution using (a) rays, (b) six, (c) two and (d) seven parameters

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Fig. 6

Solution subsets (a) real F and (b) interval F

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Fig. 7

Solution set using closed half-planes

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Fig. 8

Solution and minimum norm solution sets using (a) six parameters and (b) two parameters

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Fig. 9

Minimum norm solution set using eight independent parameters

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Fig. 10

Solution and minimum norm solution sets for interval F

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Fig. 11

Solution subsets (a) real F and (b) interval F

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